124 PART II. SOME IMPORTANT METHODOLOGICAL TERMS. 



Clifford lucidly sums up the practical difficulties involved in attaining 

 to complete exactness. We make no apology for quoting him at length: 

 "When a student is first introduced to those sciences which have come 

 under the dominion of mathematics, a new and wonderful aspect of Nature 

 bursts upon his view. He has been accustomed to regard things as essen- 

 tially more or less vague. All the facts that he has hitherto known have 

 been expressed qualitatively, with a little allowance for error on either 

 side. Things which are let go fall to the ground. A very observant man 

 may know also that they fall faster as they go along. But our student 

 is shown that, after falling for one second in a vacuum, a body is going 

 at the rate of thirty-two feet per second, that after falling for two seconds 

 it is going twice as fast, after going two and a half seconds two and a 

 half times as fast. If he makes the experiment, and finds a single inch 

 per second too much or too little in the rate, one of two things must 

 have happened: either the law of falling bodies has been wrongly stated,, 

 or the experiment is not accurate there is some mistake. He finds reason 

 to think that the latter is always the case; the more carefully he goes 

 to work, the more of the error turns out to belong to the experiment. 

 Again, he may know that water consists of two gases, oxygen and hydrogen, 

 combined; but he now learns that two pints of steam at a temperature 

 of 150 centigrade will always make two pints of hydrogen and one pint 

 of oxygen at the same temperature, all of them being pressed as much 

 as the atmosphere is pressed. If he makes the experiment, and gets rather 

 more or less than a pint of oxygen, is the law disproved? No; the steam 

 was impure, or there was some mistake. Myriads of analyses attest the 

 law of combining volumes; the more carefully they are made, the more 

 nearly they coincide with it. The aspects of the faces of a crystal are 

 connected together by a geometrical law, by which, four of them being" 

 given, the rest can be found. The place of a planet at a given time is 

 calculated by the law of gravitation; if it is half a second wrong, the fault 

 is in the instrument, the observer, the clock, or the law; now, the more 

 observations are made, the more of this fault is brought home to the 

 instrument, the observer, and the clock. . . . 



At this point we have to make a very important distinction. There are 

 two ways in which a law may be inaccurate. The first way is exemplified 

 by that law of Galileo which I mentioned just now: that a body falling 

 in vacuo acquires equal increase in velocity in equal times. No matter 

 how many feet per second it is going, after an interval of a second it 

 will be going thirty-two more feet per second. We now know that this 

 rate of increase is not exactly the same at different heights, that it depends 

 upon the distance of the body from the centre of the earth; so that the 

 law is only approximate ; instead of the increase of velocity being exactly 

 equal in equal times, it itself increases very slowly as the body falls. 

 We know also that this variation of the law from the truth is too small 

 to be perceived by direct observation on the change of velocity. But 

 suppose we have invented means for observing this, and have verified 

 that --the increase of velocity is inversely as the squared distance from 

 the earth's centre. Still the law is not accurate; for the earth does not 

 attract accurately towards her centre, and the direction of attraction is 

 continually varying with the motion of the sea; the body will not even 

 fall in a straight line. The sun and the planets, too, especially the moon, 

 will produce deviations; yet the sum of all these errors will escape our 

 new process of observation by being a great deal smaller than the neces- 

 sary errors of that observation. But when these again have been allowed 

 for, there is still the influence of the stars. In this case, however, we 

 only give up one exact law for another. It may still be held that if the 

 effect of every particle of matter in the universe on the falling body were 

 calculated according to the law of gravitation, the body would move exactly 

 as this calculation required. And if it were objected that the body must 

 be s}ightly magnetic or diamagnetic, while there are magnets not an infinite 

 way off; that a very minute repulsion, even at sensible distances, accom- 



